Econometrica: Apr 1977, Volume 45, Issue 3

A Note on Trend Removal Methods: The Case of Polynomial Regression versus Variate Differencing

https://doi.org/0012-9682(197704)45:3<737:ANOTRM>2.0.CO;2-D
p. 737-744

J. Keith Ord, Jack C. Hayya, K. Hung Chan

This paper deals with the theoretical development of some aspects of the trend removal problem. The objective is to show the difference between the two most popular trend removal methods: first differences and linear least squares regression. On the one hand, we show that if first differences are used to eliminate a linear trend, the series of residuals would be stationary but would not be white noises as they contain a first lag autocorrelation of -0.50. Furthermore, the spectral density function (SDF) of these residuals relative to that of a white noise series would be exaggerated at the high frequency portion and attenuated at the low frequency portion. On the other hand, we show that the regression residuals from the linear detrending of a random walk series would contain large positive autocorrelations in the first few lags. Relative to that of white noises, the SDF of the regression residuals would be exaggerated at the low frequency portion and attenuated at the high frequency portion.

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